I wanted to continue a little bit with the idea of Jacob Burkhardt on the crises in history. I had mentioned previously two different styles of visualization or a geometric models of history where you could imagine some multi-dimensional space in which a point represents a kind of historical state at a given time, and then another point would represent historical state of the same system at another time, and as time carries on, these points move along a supposedly continuous, or most of the time continuous curve in this state space, the geometrical model of history or culture or society or ethnography. And every once in a while there might be a discrete jump, at which the state suddenly disappears at one point and more or less simultaneously or an instant later reappears at another point farther away, and this is the kind of thing that Burkhardt called crises in history. This is exactly a century before my arrival in Santa Cruz, another crisis in history. So I'm reinterpreting Burkhardt in the mathematical style. It was not born until a little bit after his time, in the birth of Chaos Theory around 1880. So in the mathematical theory called Dynamics, aka Chaos Theory, we have this word we used, Paul used in his lecture a week or so ago, bifurcations. And this is an example of the bifurcations that in dynamical systems theory the mathematicians had tried to classify all possible bifurcations in the context of a mathematical model properly defined, which could be used as a cognitive strategy to understand history. There's never going to be a completely described mathematical model for history, of course, it's just a cognitive strategy. And the study of the mathematical models carried out primarily with computer simulation and visualization by computer graphics, they give us experience in some simple models. This experience then can be applied to a more complex dynamical system as an aid to understand, to grok what is going on.
So mathematicians have classified these bifurcations in three types: There are solo bifurcations, explosive bifurcations, and catastrophic bifurcations. The bifurcations are viewed not in this picture, called the state space, but in a slightly larger picture, called the bifurcation diagram, in which you imagine on one face, or some of the dimensions, are a copy of the state space here, and some other dimensions, visualized through just this one, are a representation of so-called parameters, for example, if you're heating up water then the amount of heat under the pot may be a parameter, and if you turn it up then you get the different kind of boiling or heating than if you turn it down. In the state space, now, it's supposed, in this context of a dynamical system with parameters, if you lieave the parameters fixed, then the dynamical system evolves to a kind of a steady state or eventual dynamical equilibrium called an attractor. So if nobody changed the heart of the flame under the pot, then this trajectory wouldn't have any jumps and would just eventually settle down in a certain configuration of boiling or simmering as the case may be, and that's called an attractor. But when parameters are changed, the dynamical rules are changed, the progress of each and every trajectory is changed, and its eventual state, its long-run behavior of the attractors, they will change also. So in the technical context of dynamical systems theory, bifurcation refers to something that happens to the configuration of attractors in the state space as the parameters are changed. So this parameter might not be timed, but the parameter might be changed in the course of time -- for example, if every once in a while you go in the kitchen and a little bit increase or decrease the amount of heat under the pot because you don't think its boiling fast enough, or its about to boil over -- as the parameters are changed, they might be changed with time gradually, but for each of these parameters there's a certain configuration of attractors, like stars in the sky, constellations, and one thing that might happen is one suddenly disappears. So for the values of the parameter to the left, there are two different attractors, and for the values of the parameter to the right, there is only attractor. And this is an example of the third type of bifurcation called catastrophic bifurcation, when an attractor suddenly appears or disappears out of the blue.
So that would be a mathematical model useable as a cognitive strategy to understand Burkhardt's idea of crises in history. Now he had published in 1968 just a short article or a chapter, transcription of a lecture, called "Crises in History," and he gave there a number of examples, but they're not developed in detail. Some of them are, for example, Alexander's conquest, the time of Constantine, the creation of the Byzantine Empire, Muhammed and the explosion of Islam across the deserts of Africa, and the Renaissance. So he gives a longer list, especially including modern history, but Burkhardt, as a historian of art and architecture, he viewed these things not as a mathematical model, but from the perspective that the characteristics of a certain attractor -- that is to say dynamical equilibrium situation of a society -- would be understandable from art historical styles. From the styles of artists you get a feel for the dynamics of the underlying society, and then there would be some kind of bifurcation, then you would see a sudden change in those styles. Burkhardt was born into a leading family in Basel, and I lived in Basel for a while, it's a really beautiful city, kind of Medieval city, every house a historical monument. And in the center of the city you have the oldest part, the oldest town. We used to walk there regularly because the only Chinese restaurant in town was there. As you walk from outside in, the dates on the houses are getting lower and lower, until you come to the town of the founding of Basel in the 13th century, in the middle, and outside you have Renaissance architecture. So here was Burkhardt, professor of the history of art and the history of architecture, walking around this town all the time. He knew the dates of every house, as if it were written on the front, like Santa Cruz Historical Society. And he observed, even in childhood, that as you walk around the city you are seeing a gradual or continuous progression of time, like time travel, in the architecture. But nevertheless there are hidden streets which are sort of arranged in wiggly concentric circles around the Chinese restaurant -- certain streets would be the boundary between radically different styles. So just growing there and living there you get a feeling for crises in history as represented in the history of architecture, in a sharp change of architectural styles in a single year before and after which there was a gradually shifting or even constant style of architecture over a period of centuries.
Well, explosive bifurcation is one in which an attractor becomes much larger. I would say, for example, the bifurcation of Alexander's conquest, listed by Burkhardt in his list of exemplary crises in history, that here you have a time when the former government changed so that the Greek states were united by conquest and at the same time with the battle of Granicus when Darius fled and Darius' Generals stabbed him and then Alex caught him and said, "Only a king can kill a king," and in the end he impaled him on spikes. At that moment there was this expansion of the culture of one culture from a smaller geographical area over a huge one; the greatest such explosion of the power of a single person in history. So maybe that would be a historical example of an explosive bifurcation that has different character than something like the Renaissance which was catastrophe in which the architectural style as it were over a large geographical area, changed overnight.
History unroles behind us. It's not that we're going forward in history. So last time I mentioned under the heading "Geometrical Models of Society" that we might take the dimensions of space to be one, two, or three dimensions of space, or even zero. The point has zero dimension. That's imagining an entire country to be a single point. And there I said, "Well, the time could be taken as zero, one or two dimensional." What do we mean by two-dimensional time? Because we have time as in a chronology, as seen by us now. And they would have time in a chronology of the same span of time as seen ten years later. Those would be totally different chronologies. So we have the time viewed and the time of the viewer, and that would be one way to invoke two-dimensional time in the geometrical model of history. For example, we (?) draw chronographs, that means we have a space spread out this way, as for example Athens, Alexandria, Byzantium, Bagdad, and time going down this way. Let's say that this chronology is as seen by us. Now if we look at one of these books, for example here there is a map of Alexandria, which I'll show you in a minute, great big map of Alexandria drawn in the Middle Ages, from a book which if it contained a chronology or a chronograph it would be a completely different one, we'd write that one on a separate piece of paper and put it on top, stacking these up as it were like a deck of cards, putting the dates of the drawing of the chronograph in temporal order, and then the time of viewing the chronology of the past would then represent a third dimension coming out this way, which is also time. Viewing time and time viewed.